[Proof of Corollary REF  ]
Let \(\beta _n(\alpha )\beta ^{\infty , 1}_n(\alpha )\)  be the non-private trade-off between type I and type II error probabilities (i.e., \(Z^n = X^n\) ). According to Chernoff-Stein lemma (see, e.g., [1]}), we have
\(\lim _{n\rightarrow \infty }\frac{1}{n}\log \beta _n(\alpha ) = -D_{\mathsf {KL}}(P_0\Vert P_1).\) 
